The situation is the following: Let $V$ be a Lie algebra of finite dimension, say n, and let be the graded commutative algebra $(\bigwedge^{\bullet}V)^*:=(\bigoplus_{k=0}^n\bigwedge^kV)^*$ and define $d:V^*\rightarrow (V\wedge V)^*$ by $f(-)\mapsto f([-,-])$. The task is now to show that this extends to a differential on $(\bigwedge^{\bullet}V)^*$, which means that we get a commutative differential graded algebra. Therefore we have to show that the following satiesfies $d$ the Leibniz rule $d(\alpha\wedge\beta)=(d\alpha)\wedge\beta+(-1)^k\alpha\wedge d\beta$. The $d$ on $(\bigwedge^{\bullet}V)^*$ should be the following:
$$d\alpha(v_1\wedge\cdots\wedge v_{n+1})=\sum_{1\leq i<j\leq n+1}{(-1)^{i+j-1}\alpha([v_i,v_j]\wedge v_1\wedge\cdots\wedge v_{i-1}\wedge v_{i+1}\wedge\cdots\wedge v_{j-1}\wedge v_{j+1}\wedge\cdots\wedge v_{n+1})}$$
I have some problems with the computations of this, can someone help me to solve this question? Thanks.